Orderability and the Weinstein conjecture (Q2786461)

One of the famous conjectures in contact geometry is the so-called Weinstein conjecture [\textit{A. Weinstein}, J. Differ. Equations 33, 353--358 (1979; Zbl 0388.58020)] which asserts for a closed coorientable contact manifold \((\Sigma, \xi)\) that any supporting contact form admits a periodic Reeb orbit. \textit{Y. Eliashberg} and \textit{L. Polterovich} [Geom. Funct. Anal. 10, No. 6, 1448--1476 (2000; Zbl 0986.53036)] introduced the notion of orderability of contact manifolds. Denote by \(\text{Cont}_0 (\Sigma, \xi)\) the group of contactomorphims of \((\Sigma, \xi)\) which are contact isotopic to the identity, and by \(\widetilde{\text{Cont}}_0 (\Sigma, \xi)\) its universal cover. Then, the authors prove that the Weinstein conjecture holds for any contact manifold \((\Sigma, \xi)\) for which \(\text{Cont}_0 (\Sigma, \xi)\) is non-orderable; if, in addition, \(\widetilde{\text{Cont}}_0 (\Sigma, \xi)\) is non-orderable, then every supporting contact form admits a contractible closed Reeb orbit.NEWLINENEWLINEThe authors also establish a link between orderable and hypertight contact manifolds, and prove a conjecture by \textit{S. Sandon} [Geom. Dedicata 165, 95--110 (2013; Zbl 1287.53067)] for certain contact manifolds.